doi:10.1038/nature08692 LETTERSli.mit.edu/Archive/Papers/10/Yu10.pdfaref1122 ghi1 123 ,f11242 ghi2...

LETTERS

Strong crystal size effect on deformation twinningQian Yu1, Zhi-Wei Shan1,2, Ju Li3, Xiaoxu Huang4, Lin Xiao1, Jun Sun1 & Evan Ma1,5

Deformation twinning1–6 in crystals is a highly coherent inelasticshearing process that controls the mechanical behaviour of manymaterials, but its origin and spatio-temporal features are shroudedin mystery. Using micro-compression and in situ nano-compressionexperiments, here we find that the stress required for deforma-tion twinning increases drastically with decreasing sample size ofa titanium alloy single crystal7,8, until the sample size is reducedto one micrometre, below which the deformation twinning isentirely replaced by less correlated, ordinary dislocation plasticity.Accompanying the transition in deformation mechanism, themaximum flow stress of the submicrometre-sized pillars wasobserved to saturate at a value close to titanium’s ideal strength9,10.We develop a ‘stimulated slip’ model to explain the strong sizedependence of deformation twinning. The sample size in transi-tion is relatively large and easily accessible in experiments, makingour understanding of size dependence11–17 relevant for applications.

Explosive growth in the use of small-volume materials such assubmicrometre-scale single crystals is driving the exploration ofsample-size-dependent mechanical behaviour15–17. At room temper-ature, the two major mechanisms responsible for the plasticity ofmaterials are ordinary dislocation plasticity (ODP) and deformationtwinning. For polycrystalline materials, it has been well-establishedthat regardless of the major deformation mechanism, the apparentstrength and the grain size (dg) follows a Hall–Petch-type behaviour:s 5 s01kdg

2a, where s is the flow strength, s0 and k are size-independent constants and a is an exponent typically between 0.5and 1. The difference is that the Hall–Petch slope for deformation-twinning-mediated plasticity (kDT) is much larger (up to ten times)than that for ODP (kODP)18–20. In recent years, size effects on theplasticity of surface-confined single crystals have drawn considerableattention15–17. It has been demonstrated that the sample dimension dstrongly influences the ODP-mediated deformation and hence theapparent strength15–17, which also follows a power-law behaviour.However, it is yet unknown if deformation-twinning-mediatedplasticity would be strongly sample-size-dependent. If so, and ifkDT ? kODP, the power-law scaling for deformation-twinning-mediated plasticity must break down at a much larger sample sizethan that for ODP-mediated plasticity, because the ideal strength ofthe crystal9 imposes an upper bound on the flow stress—see Sup-plementary Fig. 1.

We chose a single-crystal Ti alloy as our model system. Deform-ation twinning is known to play a key part in the deformation ofhexagonal-close-packed metals7,8 owing to the limited number ofdislocation slip systems. For compression of bulk Ti and a-Ti alloysalong the c-axis8,21–24, deformation twinning is the geometricallyfavoured deformation mode and the reported8,21–24 twinning systemsare f11�222g �11�1123h i, f11�224g �22�2243h i and f10�111g �11012h i. In conventionalsamples, the twin lamellae in Ti and its alloys have thicknessesranging from 0.1 to 10 mm and lengths of the order of several

micrometres7,8,21–24. It is then interesting to see what will happenwhen the sample dimensions are reduced to the same scale or evensmaller.

We used a bulk square Ti–5 at.% Al single crystal as the startingmaterial, from which all the samples in this study were cut. Theexperimental details are described in the Methods and in theSupplementary Information. Supplementary Fig. 2 shows the beha-viour of the bulk single crystal under [0001] compression. Profusedeformation twinning is seen, in agreement with the literature24. Inmicro-compression tests of pillars with d $ 1.0 mm, almost all thedeformed samples showed obvious shearing traces on the surfaces.Examples are shown in Fig. 1a and b. A trace analysis of thed 5 1.0 mm micro-pillar showed that the shearing occurred on the{11�222} plane, which is a common twinning plane in hexagonal-close-packed Ti and its alloys. Electron backscatter diffraction (EBSD)analysis of these deformed pillars provides evidence that deformationtwinning indeed occurred during compression. Figure 1c and d com-pare the pole figures of the d 5 8.0 mm pillar, before and after thedeformation. We observed nearly perfect {0001} reflection in themicro-pillar before loading, but after deformation several new orien-tations that are far from the initial orientation appeared. We identifythese new orientations as due to deformation twinning on two typesof twinning systems, f11�222g �11�1123h i, and f10�111g �11012h i, as indicatedin Fig. 1d. Similar EBSD results were obtained for the d 5 1.0 mmsample as well. Rapid deformation twin growth is also corroboratedby the obvious strain bursts in the stress–strain curves; see Fig. 2a.Transmission electron microscopy (TEM) micrographs of thedeformed d 5 8 mm pillar, taken from its vertical cross-section cutusing focused ion beam (FIB) micromachining (Supplementary Fig. 3),are shown in Supplementary Figs 4 and 5: deformation twins areobserved inside tangles of curved dislocations. The twins with smallsizes—that is, twin embryos—exhibit convex lens shapes (Supplemen-tary Fig. 5), in contrast to the straight profile of grown twin bands(Supplementary Fig. 2).

We noticed remarkable changes in deformation behaviour ford 5 0.7 mm and 0.4 mm. As shown in Fig. 3a and b, in these submicro-metre pillars plastic deformation is concentrated on the top part ofthe pillar, resulting in a ‘mushroom’ shape. The load-displacementcurve of this sample is given in Fig. 2b. In Fig. 2b we observe con-tinuous plastic flow without any major strain burst. This differencefrom Fig. 2a suggests a dramatic change in the deformation mode ascompared with the larger, micrometre-sized pillars.

We also conducted compression tests in situ inside a TEM for a0.25-mm-diameter cylindrical pillar, to reveal the entire dynamicdeformation process in these submicrometre pillars (see movie inthe Supplementary Information). The still images comparing thestructure before and after testing are displayed in Fig. 3c and d.The corresponding load-displacement curve is in Fig. 2c. Here again,we observe the development of mushroom-like sample morphology

1Center for Advancing Materials Performance from the Nanoscale (CAMP-Nano), State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an, 710049,China. 2Hysitron Incorporated, 10025 Valley View Road, Minneapolis, Minnesota 55344, USA. 3Department of Materials Science and Engineering, University of Pennsylvania,Philadelphia, Pennsylvania 19104, USA. 4Danish-Chinese Center for Nanometals, Materials Research Division, Risø National Laboratory for Sustainable Energy, Technical University ofDenmark, DK-4000 Roskilde, Denmark. 5Department of Materials Science and Engineering, The Johns Hopkins University, Baltimore, Maryland 21218, USA.

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and continuous plastic flow. TEM micrographs in Fig. 3c show thatbefore the test, the density of dislocations is very low, except at the topof the pillar where a few dislocation lines were left from FIB damage(our TEM examinations indicate that the damage layer is only a fewnanometres thick and should not affect the mechanism transition atthe size scale of dc < 1 mm). After the compression test, a high densityof tangled dislocations is observed; see Fig. 3d. There is no sign ofdeformation twins from either the images or the diffraction patterns.The movie in the Supplementary Information, when correlated withFig. 2c, reveals that during the first and second loading, only a fewdislocations propagated from the contact surface into the pillar. Thedislocation activities became intense only during the third loading, atwhich point the load reached 105 mN (the corresponding contactstress is ,2 GPa, when dividing the load by the contact area). Thecontinuous generation, multiplication and considerable accumula-tion of dislocations (see also Fig. 3d) are consistent with Fig. 2b andFig. 2c: these submicrometre pillars exhibit smooth load-displacementcurves in lieu of the large strain bursts corresponding to rapid deforma-tion twin growth (Fig. 2a).

These results clearly demonstrate that deformation twinning isstrongly dependent on the sample size. In micrometre-sized pillars,deformation twinning is still the dominant mode of plastic deforma-tion, as in bulk-sized samples8,21–24. As shown in Fig. 2d for the pillarswith d $ 1 mm, the strength (red symbols) and d (here the strengthvalues are defined as the true stress measured at the twinning events,that is, the bursts in Fig. 2a) clearly follow a Hall–Petch-type rela-tionship with a < 1. We note that this strong size effect on deforma-tion twinning is intrinsic to the material, because sample tapering isabsent and the very thin FIB damage layer on the surface should benegligible in these relatively large samples.

At submicrometre sample sizes, in contrast, deformation twinningis suppressed, giving way to ODP only, even though our loadingdirection is favourable for deformation twinning. In addition, strain

bursts were not observed and the pillars were deformed into a mush-room-like geometry. For these submicrometre samples, we havedefined the maximum flow stress as the peak load divided by the areaof the narrowest cross-section pdnarrowest

2/4 of the mushroomed pil-lars (see inset in Fig. 2d) and plotted it in Fig. 2d. It is interesting tofind that the maximum flow stress saturated at close to the material’sideal strength10, which is 2.8–4.9 GPa in pure shear for Ti9.

We wondered why deformation twinning follows a power-lawscaling on d, and why kDT? kODP. Here we present a ‘stimulatedslip’ model to explain the size dependence of deformation twinning.In contrast to ODP, where inelastic shear activities are randomlydispersed among slip planes, deformation twinning is characterizedby perfectly correlated layer-by-layer shearing: all slip (twinningdislocations) on atomically adjacent parallel planes must have thesame Burgers vector. We posit that deformation twinning in relationto ODP is akin to what laser (light amplification by stimulatedemission of radiation)25 is to normal light, in the sense of slipcoherence.

Here we consider a cuboidal crystal with size d and a total disloca-tion density r (in units of m22). These stored lattice dislocations maypenetrate a twinning slip plane n of area d2 (see Fig. 4). We assumethat a small fraction Ppromoter of the dislocations that penetrate planen can play the role of ‘promoters’ (for example screw dislocations invarious pole mechanisms2,6,26–28), that ‘stimulate’ slip of the samecharacter to occur also on the next atomic plane. Specifically, weassume that when a moving slip front (glide dislocation with twin-ning Burgers vector) on plane n hits and wraps around a promoter,inelastic shear of the same character ‘infects’ plane n 1 1; and onceinfected anywhere on plane n 1 1, we assume this stimulated twin-ning slip will propagate and cover the entire plane n 1 1 as well, if thestress is high enough to drive it pass the forest dislocation obstacles,thickening the deformation twinning. The layer-to-layer ‘infectionprobability’ is simply

(112–2)

0.5 μm

5.0 μm

a c Before test

d After testb

x: [101–0]

Min12345

Max

0306090

120150

180.38

x: [101–0]

T1

T1

T2

T2

{0001}

{0001}

y: [1–21

–0]

y: [1–21

–0]

Min12345

Max

01020304050

59.37

Figure 1 | Scanning electron microscopy imagesof the deformed micropillars and EBSD polefigure. a, d 5 8.0 mm. b, d 5 1.0 mm. c, d, EBSDpole figure of the d 5 8.0 mm pillar before (c) andafter (d) compression. T1: f11�222g �11�1123h i and T2:f10�111g �11012h idenote the twin type. T1 and T2

have misorientation of 64u/,1�1100w and 57u/,2�11�110., respectively, with respect to the initialorientation. (Half-width 10u; cluster size 5u.)

LETTERS NATURE | Vol 463 | 21 January 2010


Pinfection < d2rPpromoter (1)

Deformation twinning keeps thickening when

1 < Pinfection (2)

and the critical dislocation density necessary for the sudden transi-tion to perfect slip coherence is thus

rc < d22Ppromoter21

(3)

From the Taylor hardening model29

s 5 s0 1 kEbr1/2(4)

where E is the Young’s modulus, b is the Burgers vector length, k is adimensionless constant of order 1, s0 contains contributions fromlattice friction, solute strengthening, and so on, we have the twinningstress

sDT 5 s0 1 (kEbPpromoter21/2)d21 5 s0 1 kDTd21

(5)

The best fit in Fig. 2d gives a 5 0.976, consistent with equation (5)above. The relatively large deformation twinning Hall–Petch slopekDT (see also Supplementary Table 1 and ref. 18) can be explained by a

small Ppromoter < 1022, which is reasonable because only a smallsubset of the stored dislocations may be screw poles of the right typefor promoting twinning slip. A full discussion of this model is pro-vided in the Supplementary Information.

A large kDT in sDT 5 s0 1 kDTd2a drives early deformation-twinning abdication (dc < 1 mm), because as sDT approaches theideal strength, there will be profuse dislocation nucleation, leadingto high ODP strain rates. We note that as the pillars enter the sub-micrometre regime, the nucleation of ODP dislocations is increas-ingly assisted by the roughness left during FIB processing at thecontact interface, and the now-very-large near-surface regions30.

We conclude that deformation twinning is an intensely collective,‘stimulated slip’ phenomenon, analogous to ‘stimulated emission’ inlaser theory25. Stimulated slip from plane n to plane n 1 1 is catalysed bypromoter defects, such as screw dislocation poles2,6,26–28. Dual require-ments must be met for deformation twinning: the stress needs to be highenough to drive a twinning dislocation to sweep plane n of area d2,cutting across forest obstacles; and of these forest obstacles, a sufficientproportion must be able to promote slip from plane n to n 1 1. Thesmaller d is, the more weakly two adjacent slip planes are effectivelycoupled by threading screw pole dislocations, and the more difficult it isfor deformation twinning . The large Hall–Petch slope kDT implies asmall promoter fraction Ppromoter among the stored bulk dislocations,leading to a very large dc, observed to be ,1mm, at which Hall–Petchbehaviour breaks down and transition to incoherent ODP occurs.

a

dnarrowest

c d

b2.8

2.4

2.0

1.6

1.2

0.8

0.4

0.0

2.5

2.0

1.5

1.0

0.5

0

0.20

0.15

0.10

0.05

0.00

–0.05

5.5

4.5

3.5

2.8

2.6

2.4

2.2

2.0

1.8

0 4 8 12 16 20 24 0 50 100 150 200 250 300

0 20 40 60 80 100 0 1 2 3 4

8.0 μm 5.0 μm2.0 μm1.0 μm

3.0 μm1.2 μm

0.4 μm

0.7 μm

Test 1

Test 2

Test 3

Test 4

Test 5

8.03.0 1.2 0.7 0.4 0.26 0.24

5.0 2.0 1.0 0.25

Eng

inee

ring

stre

ss (G

Pa)

Flow

str

ess

(GP

a)Lo

ad (m

N)

Load

(mN

)

Engineering strain (%) Displacement (nm)

Displacement (nm) d–1 (μm–1)

d (μm)

Figure 2 | Mechanical data of the testedsamples. a, The stress–strain curves ofmicropillars with decreasing side length, d from8.0 to 1.0mm. b, The load-displacement curves ofsubmicrometre pillars. c, The load-displacementcurves of a submicrometre cylindrical pillar with0.25mm diameter, in five consecutiveload–unload steps during in situ testing inside aTEM (see movie in the SupplementaryInformation). The negative forces at the end ofthe unloading segments are due to adhesionbetween diamond tip and the pillars. d, Flowstress measured for the pillars versus d. We usethe narrowest cross-section pdnarrowest

2/4 tocalculate the flow stress. The error bars are twostandard deviations.

NATURE | Vol 463 | 21 January 2010 LETTERS


This size is of considerable technological importance. When thesurface-bound contiguous crystal is reduced to the submicrometrescale, the effective interlayer coupling responsible for coherent ‘sti-mulated slip’ becomes so weak that deformation twinning is nolonger triggered by bulk promoters, and incoherent ODP at highstress is sufficient to match the imposed strain rate. We note thatthe discussions above are based on a bulk promoter population,which scales with the volume. Deformation twinning proficiencymay go up again at very small crystal sizes such as 10 nm (ref. 1),when the most effective promoters for stimulated slip may be grainboundaries and surfaces instead of screw dislocations, on the basis ofan area-to-volume ratio argument.

METHODS SUMMARYThe orientation of the single crystal a-Ti alloy was determined using the Laue

back reflection method. The pillar samples oriented along the c-axis for com-

pression testing were prepared using FIB. Compression testing was performed

using an MTS Nanoindenter XP and a Hysitron TEM PicoIndenter, respectively.

Received 13 July; accepted 17 November 2009.

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2. Christian, J. W. & Mahajan,S. Deformation twinning. Prog.Mater. Sci.39, 1–157 (1995).3. Ogata, S., Li, J. & Yip, S. Energy landscape of deformation twinning in bcc and fcc

metals. Phys. Rev. B 71, 224102 (2005).4. Wu, X. L. & Zhu, Y. T. Inverse grain-size effect on twinning in nanocrystalline Ni.

Phys. Rev. Lett. 101, 025503 (2008).5. Warner, D. H., Curtin, W. A. & Qu, S. Rate dependence of crack-tip processes

predicts twinning trends in f.c.c. metals. Nature Mater. 6, 876–881 (2007).6. Niewczas, M. in Dislocations in Solids Vol. 13 (eds Nabarro, F. R. N. & Hirth, J. P.)

263–364 (Elsevier, 2007).7. Song, S. G. & Gray, G. T. Structural interpretation of the nucleation and growth of

deformation twins in Zr and Ti. Acta Metall. Mater. 43, 2325–2350 (1995).8. Williams, J. C., Baggerly, R. G. & Paton, N. E. Deformation behavior of HCP Ti-Al

alloy single crystals. Metall. Mater. Trans. A 33, 837–850 (2002).9. Ogata, S., Li, J., Hirosaki, N., Shibutani, Y. & Yip, S. Ideal shear strain of metals and

ceramics. Phys. Rev. B 70, 104104 (2004).10. Suresh, S. & Li, J. Deformation of the ultra-strong. Nature 456, 716–717 (2008).11. Lu, L., Chen, X., Huang, X. & Lu, K. Revealing the maximum strength in

nanotwinned copper. Science 323, 607–610 (2009).12. Argon, A. S. & Yip, S. The strongest size. Phil. Mag. Lett. 86, 713–720 (2006).13. Schiotz, J. & Jacobsen, K. W. A maximum in the strength of nanocrystalline

copper. Science 301, 1357–1359 (2003).14. Shan, Z. W. et al. Grain boundary-mediated plasticity in nanocrystalline nickel.

Science 305, 654–657 (2004).15. Uchic, M. D., Dimiduk, D. M., Florando, J. N. & Nix, W. D. Sample dimensions

influence strength and crystal plasticity. Science 305, 986–989 (2004).16. Shan, Z. W., Mishra, R. K., Asif, S. A. S., Warren, O. L. & Minor, A. M. Mechanical

annealing and source-limited deformation in submicrometre-diameter Nicrystals. Nature Mater. 7, 115–119 (2008).

17. Greer, J. R., Oliver, W. C. & Nix, W. D. Size dependence of mechanical propertiesof gold at the micron scale in the absence of strain gradients. Acta Mater. 53,1821–1830 (2005).

18. Meyers, M. A., Vohringer, O. & Lubarda, V. A. The onset of twinning in metals: aconstitutive description. Acta Mater. 49, 4025–4039 (2001).

19. Stanford, N., Carlson, U. & Barnett, M. R. Deformation twinning and the Hall-Petchrelation in commercial purity Ti. Metall. Mater. Trans. A 39, 934–944 (2008).

20. El-Danaf, E., Kalidindi, S. R. & Doherty, R. D. Influence of grain size and stacking-fault energy on deformation twinning in fcc metals. Metall. Mater. Trans. A 30,1223–1233 (1999).

21. Paton, N. E. & Backofen, W. A. Plastic deformation of titanium at elevatedtemperatures. Metall. Trans. 1, 2839–2847 (1970).

22. Akhtar, A. Basal slip and twinning in alpha-titanium single-crystals. Metall.Trans. A 6, 1105–1113 (1975).

23. Yoo, M. H. Twinning and mechanical behavior of titanium aluminides and otherintermetallics. Intermetallics 6, 597–602 (1998).

24. Xiao, L. Twinning behavior in the Ti-5 at.% Al single crystals during cyclic loadingalong. Mater. Sci. Eng. A 394, 168–175 (2005).

25. Svelto, O. Principles of Lasers (Springer, 1998).26. Cottrell, A. H. & Bilby, B. A. A mechanism for the growth of deformation twins in

crystals. Phil. Mag. 42, 573–581 (1951).27. Niewczas, M. & Saada, G. Twinning nucleation in Cu-8 at.% Al single crystals. Phil.

Mag. A 82, 167–191 (2002).28. Song, S. G. & Gray, G. T. Double dislocation pole model for deformation twinning

in fcc lattices. Phil. Mag. A 71, 661–670 (1995).29. El-Azab, A. The statistical mechanics of strain-hardened metals. Science 320,

1729–1730 (2008).30. Zhu, T., Li, J., Samanta, A., Leach, A. & Gall, K. Temperature and strain-rate

dependence of surface dislocation nucleation. Phys. Rev. Lett. 100, 025502 (2008).

Supplementary Information is linked to the online version of the paper atwww.nature.com/nature.

Acknowledgements We thank Q. Liu for help with EBSD experiments. This workwas supported by grants from the NSFC (50671077, 50720145101, 50831004 and50925104), the 973 Program of China (2004CB619303, 2007CB613804 and2010CB613003) and the 111 Project of China (B06025). J.L. was supported by ONRgrant N00014-05-1-0504, NSF grant CMMI-0728069, MRSEC grantDMR-0520020 and AFOSR grant FA9550-08-1-0325. X.H. was supported by theDanish National Research Foundation. The in situ TEM work was performed at theNational Center for Electron Microscopy, Lawrence Berkeley Laboratory, which issupported by the US Department of Energy under contract DE-AC02-05CH11231.

Author Contributions Q.Y. and Z.-W.S. carried out the experiments, J.L.constructed the model, X.H. interpreted the EBSD results, L.X. supervised thesample selection, J.S. designed the project, J.L. and E.M. wrote the paper. Allauthors contributed to the discussions.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Correspondence and requests for materials should be addressed to J.S.([email protected]) or J.L. ([email protected]).

Crystal size d

Plane n

Plane n–1

Plane n–2

…

Plane n+1

Slippedalready

Needto slip

Bulk dislocation density, ρ (m–2)

Incomingslip front

on plane n

Stimulatedslip onplane n+1

…

Figure 4 | Schematic of the ‘stimulated slip’ model. Thickening ofdeformation twin by ‘stimulated slip’ occurs when an incoming slip front(twinning glide dislocation) hits bulk screw dislocation ‘promoters’, a subsetof the bulk forest dislocation population.

0.5 μm

0.2 μm

Before test

g = [0002]

200 nm [011–

0]

[011–

0]

a

b

200 nm

g = [0002]

After test

c

d

Figure 3 | Electron microscopy images of the tested samples. a, b, SEMimages of the deformed d 5 0.7 mm (a) and d 5 0.4 mm (b) pillars.c, d, Centred dark-field TEM images with diffraction pattern (insets) of the0.25-mm-diameter pillar before (c) and after (d) the in situ compression test.The beam direction was [01�110] and the reflection vector g 5 [0002].

LETTERS NATURE | Vol 463 | 21 January 2010



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mailto:[email protected]

mailto:[email protected]

A simple schematic that summarizes the main finding of our paper is Fig. S1.

FIG S1: Comparison of Hall-Petch scaling and Hall-Petch breakdown between ordinary

dislocation plasticity (ODP, black curve)1-3 and deformation twinning (DT, magenta curve).

Deformation twinning is much more size aware than ordinary dislocation plasticity in BCC

and HCP metals, as indicated in Table S1 for the grain size dependence.

~ Theoretical Strength

d-α

Strength

20nm1000nm

σ=σ0+kDTd-α

σ=σ0+kODPd-α

ODP: Ordinary Dislocation Plasticity

DT: Deformation Twinning

SUPPLEMENTARY INFORMATIONdoi: 10.1038/nature08692

www.nature.com/nature 1

Table S1: Taken from Meyers, M. A., Vohringer, O., Lubarda, V. A., “The onset of twinning

in metals: A constitutive description,” Acta Mater. 49 (2001) 4025, for grain size effect on

deformation twinning.

doi: 10.1038/nature08692 SUPPLEMENTARY INFORMATION


1. Methods

The crystal orientation of the Ti-5at%Al single crystal was determined using the Laue back

reflection method. A group of micron and submicron pillar samples oriented along the c-axis

for compression testing were prepared using the Focused Ion Beam (FIB) micromachining

technique4. The samples have square cross-sections with the side length (d) in the range of

8.0 to 0.4 μm, and an aspect ratio of ~2:1. Two orthogonal faces, which were used for surface

trace analysis, were made to be parallel to the (10 1 0) plane and the (1 2 10) plane,

respectively. In addition, cylindrical pillars with a diameter of about 250 nm were fabricated

by FIB cutting, using the method described by Shan et al.5. All the samples have their long

(height) axis parallel to the [0001] direction (this orientation was confirmed to remain

unchanged after testing).

Compression testing was performed using an MTS Nanoindenter XP, outfitted with a flat-tip

diamond punch, at a nominal axial strain rate of 5×10-4/s to 1×10-4/s at room temperature. For

micro-pillars with d > 1 μm, electron backscatter diffraction (EBSD) with a step size of 50

nm in a field-emission scanning electron microscope (SEM) was used to investigate the

evolution of the microstructure (such as the orientation of the twins) in the deformed samples.

TEM samples were prepared using FIB, for microstructural observation in a JEOL 2100F

microscope. The centered dark field TEM images presented were not taken under strict

weak-beam conditions, due to the limited tilting capability available inside the holder. For

the 250 nm samples, in situ nano-compression experiments were carried out at room

temperature in a JEOL 3100 TEM operating at 300kV, employing a Hysitron TEM

PicoIndenter 5; this test directly correlates the microstructural evolution with the mechanical

data throughout the test.

Movie of the in situ compression of the 250 nm Ti-5at%Al single crystal pillar in TEM is

placed at Nature website.



2. Deformation Twinning in Bulk Ti-5%Al

FIG S2: (a) Engineering stress-strain curve and (b) TEM micrographs of Ti-5at%Al single

crystal loaded in the [0001] direction. The sample in (a) has dimensions 3 mm×3 mm×6 mm.

Note the typical dimensions and morphologies of the deformation twins in (b).



3. TEM Characterization of Deformed Ti-5%Al Micropillars

We have carried out transmission electron microscopy characterizations of the deformed

pillars. The TEM slab specimens were cut using focused ion beam. Details of the sample

preparation procedures are illustrated in Fig. S3. The results, shown below in Fig. S4 and Fig.

S5 as examples, indicate that deformation twinning indeed occurs in the midst of lattice

dislocations in d>dc pillars. A typical example showing multiple deformation twins is given in

Fig. S4A which is a centered dark field TEM image. The diffraction spots from the matrix

and those from the twins are marked with blue and green line, respectively in the

corresponding selected area diffraction pattern (Fig. S4B). The diffraction spot used for

imaging Fig.S4A is indicated by the short green arrow in Fig. S4B. From these, some of the

twins can be identified as of the {1011} 1012 type.

It is worth noting that the twins with smaller sizes, i.e. twin embryos, exhibit convex lens

shape. This is in contrast to those observed in their bulk counterpart where the twin

boundaries are usually straight and long (Fig. S2b). The deformation twin embryos are indeed

located in a region of high dislocation densities (Fig. S5). The majority of the dislocation

segments are of edge character. However, the ledges associated with the long dislocation lines

and the bowing-out ends of dislocations in the tangled dislocations are screw segments,

prerequisite for the pole mechanism. From g·b analysis, we have found dislocations with

Burgers vectors of all the three major types, <a>, <c> and <a+c>.

When d>dc (micrometer sized) pillars are deformed plastically but are cut open prior to strain

burst in the stress vs. strain curve, TEM analyses show dislocation conformation and density

similar to Fig. S5, but no twins. These results provide the experimental evidence of the prior

occurrence of ordinary dislocation plasticity (ODP, “incoherent slip”) and dislocation density

storage as a prerequisite for the initiation of deformation twinning (“coherent slip”). It is also

consistent with the other results in several respects. First, the stress vs. strain (load vs.

displacement) curves (Fig. 2a & Fig. S2a) clearly showed that considerable ODP, which gives



smooth flow, occurred prior to the strain bursts (twin initiation). Second, post mortem TEM

observations in a deformation twinning dominated sample confirmed the existence of a large

number of dislocations besides the deformation twins (Fig. S2b). Third, in situ TEM

compression tests showed the prevalence of dislocation activity in the absence of deformation

twinning (see movie at Nature website). The observed curved dislocation tangles include

many dislocation segments with screw character.



Fig. S3: Cross-section-view TEM sample preparation of a deformed pillar with o8μm diameter. (A) deformed sample; (B) A layer of Pt was deposited on top of the deformed pillar; (C) A thin slab of sample was cut with focused ion beam; (D) e-beam image of the slab prior to be cut off; (E) Sample lift out; (F) Sample was glued to an Omniprobe grid for further thinning.



Fig. S4: Deformation twins in the deformed pillar in Fig. S3. (A) is a typical dark field TEM image of deformation twins. Note the smaller twin indicated by the short red arrow exhibit a convex lens shape. (B) is the corresponding selected area diffraction pattern of (A). The blue and green lines mark the diffraction spots from the matrix and the twin, respectively. The diffraction vector used to image (A) was highlighted with the green arrow in (B).



Fig. S5: Twin embryo observed in a matrix of dislocations. (A) and (B) are bright field and centered dark field TEM images, respectively. The inset in (A) is the corresponding selected area diffraction pattern. The twin embryo (white arrow in both A and B) exhibited typical convex lens shape. The diffraction vector used for imaging was g=[0002].



4. A “Stimulated-Slip” Model for Size-Dependent Deformation Twinning

Deformation twinning (DT) is a highly correlated atomic shearing process6, which we deduce

in the model below to arise out of ODP (ordinary dislocation plasticity) but then supercedes

ODP if d > dc. In the model below, we will employ the analogy that DT versus ODP is like

what laser (Light Amplification by Stimulated Emission of Radiation) is to normal light. To

begin, let us consider a cuboidal crystal volume consisting of m slip planes, say m = 10,000.

If this crystal volume sustains a plastic shear strain of 1%, it would mean 100 planes have

slipped. If this occurred by ODP, it is likely that the 100 slipped planes are nearly randomly

dispersed among the 10,000, which means that on average ~102 planes separate one slipped

plane from the next closest slipped plane. This means the probability that the two slipped

planes are situated right next to each other is very small. If so, slip activities on the two

planes are unlikely to be strongly correlated. In contrast, if the 1% plastic strain is

accomplished by DT, then it is very likely that tens, or even the entire 100, slipped planes are

adjacent right next to each other, in blocks. In many experiments it is not difficult to see

micrometer-thick deformation twins. This means that for DT the average distance between

one slipped plane and the next closest slipped plane will be close to 1 atomic layer spacing,

whereas it is ~102 in ODP. Therefore, slip activities are much more strongly correlated in

DT than in ODP, on parallel planes at the atomic level. This has an underlying atomic

energetics reason6. What is intriguing, however, is that the propensity for DT versus ODP

also depends on the crystal volume, i.e., the size effect. Our experiments indicate that the

larger the contiguous crystal volume, the perfectly coherent DT mode is more favored over

the less collective, incoherent ODP mode.

Another intriguing fact is that while DT and ODP are both found experimentally to follow a

Hall-Petch type size-dependent strengthening behavior: σ=σ0+kd-α, DT has much larger

Hall-Petch slope than ODP: kDT >> kODP in many materials, sometimes by an order of

magnitude7, for the same α. This means DT is even more size-aware than ODP. Such rapid

rise in flow stress with shrinking size means the breakdown of DT Hall-Petch scaling relation

must occur at a much larger critical size dC than what is typically expected for ODP (~20 nm)



2,3,8. When one uses the ideal shear strength of pure Ti (calculated from density functional

theory to be 2.8-4.9 GPa9) as an upper bound for σ, and kDT measured from μm-sized Ti alloy

crystals, one anticipates a DT Hall-Petch breakdown dC from several hundred nms to 1 μm, a

much more conspicuous and experimentally accessible (and engineering-wise important)

sizescale than the dC ~ 20nm for the ODP-grain boundary sliding transition in nanocrystals 2,3,10.

Since the large dC for DT is tied to its large kDT, the key to understanding the Hall-Petch

behavior and its breakdown is answering:

(a) What physical mechanism drives σ=σ0+kd-α behavior for deformation twinning?

(b) Why is kDT so large?

We believe both (a) and (b) can be satisfactorily answered by inspecting the highly correlated,

“stimulated slip” nature of deformation twinning.

We hypothesize that when slip progresses on one atomic plane n, there can be certain

“promoter” defects such as screw dislocations, grain boundaries, and others, that can promote

slip of identical character to also occur on plane n+1, parallel and immediately adjacent to

plane n. This is termed “stimulated slip” here, analogous to the concept of “stimulated

emission” in laser theory11. The idea is that there is an enhanced probability of having slip

of the same character (same Burgers vector) to also occur on plane n+1, above and beyond

the average probability of having “spontaneous slip” on plane n+1 from ODP, if (i) the

atomic plane n is undergoing slip of this Burgers vector, and (ii) plane n intersects a promoter

defect. Classic examples of promoter defect and “stimulated slip” are various kinds of pole

mechanisms7,12-14. Although details vary, the common key element of pole mechanisms is

bulk screw dislocation(s) intersecting the twinning glide plane. The screw dislocation(s)

warps originally disjoint parallel slip planes into a continuous staircase helix, such that slip

can propagate up or down the staircase. One can also envision various kinds of rebound



mechanisms 15 occurring at the boundary of a crystal volume. For instance, in a FCC crystal

when one Shockley partial dislocation hits a grain boundary or free surface, there can be

enhanced probability that another Shockley partial emerges at the locale of the hit, but on one

plane above or below, which will then move backward and proceed to thicken the

deformation twin. These detailed mechanisms can be modeled atomistically to high

accuracy16,17. One however could proceed with the generic concept of “promoter” defects

and “stimulated slip” to obtain size-dependent DT behavior, without specifying the detailed

atomistic mechanisms.

Next, we examine the consequence of a bulk “promoter” defect population on the size scaling

of DT, specifically pole mechanisms requiring bulk screw dislocations, which are part of the

general dislocation population characterized by the total dislocation density ρ (unit 1/m2).

While boundary “promoters” such as grain boundary or free surfaces are likely to become

important at size scales of tens of nanometers and below18, we suggest that boundary

“promoters” are less important in our present experiments dealing with characteristic

dimensions near 1 μm, based on a area-to-volume ratio argument.

In Fig. 4 illustration of the main text, we presume stimulated slip has already occurred on

planes n-1, n-2, … and is sweeping on plane n. Since the size scale of the contiguous

crystal volume is d, planes n, n-1, n-2, … all have area ~d2. Here, it is important to

appreciate the way size d2 comes into the problem. Like laser, deformation twinning requires

perfect coherence in slip activities, i.e. slip on plane n, n-1, n-2, … must all have the same

Burgers vector. Suppose our contiguous crystal volume is bounded by grain boundaries,

when stimulated slip hits a grain boundary this perfect coherence must be heavily disrupted.

This is because the adjacent grain is misoriented with respect to our grain, with non-parallel,

intersecting slip planes, so slip must change character even if it wants to proceed onto the

next grain. Furthermore, grain boundaries have larger CSL-period (coincident site lattice) or

even random structures, which will also heavily disrupt the perfect slip coherence. Therefore,

the area-wise extent of perfect slip coherence is likely bounded by d2. The key to the survival

of the deformation twinning mode is whether perfect slip coherence can be sustained in the



thickness direction. In order to achieve this, the one-layer-to-next-layer “infection

probability” Pinfection must be nearly 100%. That is to say, conditional upon plane n is

slipping and plane n+1 has not yet slipped, the probability that plane n+1 will get “infected”

to slip in the same way (stimulated slip) must be nearly 1, in order to achieve runaway

propagation of perfect slip coherence in the thickness direction. Otherwise, one cannot

possibly see deformation twins microns thick in experiments, involving thousands of

consecutive atomic planes.

In our simple model, we assume that a fraction Ppromoter (0<Ppromoter<1) of the total dislocation

population act as promoter dislocations. We also assume that when the slip sweeps on

plane n, if it hits a forest dislocation that penetrates the plane, which turns out to be a

promoter, inelastic shear of the same character will infect plane n+1. Once infected

anywhere on plane n+1, of area d2, this stimulated slip will eventually propagate and cover

the entire d2 area at some speed. Thus, the one-layer-to-next-layer “infection probability” is

simply

Pinfection = d2 ρ Ppromoter . (1)

In order to trigger perfectly coherent DT, there needs to be

1 ≈ Pinfection . (2)

According to (1),(2), if one starts deformation on a well-annealed sample with low ρ, DT

must be preceded by ODP activities, working like an optically-pumped laser. The temporal

sequence would be the following. In experiments where stress is ramped up from zero, there

are always some ODP activities even at the beginning. It is just that these ODP dislocations

have high core energy and/or low mobility and/or low Schmid factors, such that their strain

rate contribution cannot match the applied strain rate at low stresses. This difference in strain

rates gets accumulated to the elastic strain and causes the stress σ to rise. However, before

the stress σ rises to high enough levels to allow sufficient ODP to accommodate all the strain

rate imposed, the ODP dislocations “catch fire and start lasing” (DT). Once the highly



correlated DT mode is established, it is fast and efficient, and can match whatever applied

strain rate that is given6. This is presumably what happens at d > dc.

The size scaling of DT stress can now be derived, since it is just the critical ODP stress

where runaway condition (see Eqs. (1) and (2) above) for the transition to perfect coherence

is satisfied. For the ODP flow stress, we have the familiar Taylor model:

σ = σ0 + κEbρ1/2 (3)

where E is the Young’s modulus, b is Burgers vector length, κ is a dimensionless constant of

order 1. σ0 contains contributions from lattice friction, solute strengthening, etc. From Eqs. (1)

and (2), we see that the sudden transition to perfect slip coherence occurs at a critical

dislocation density

ρc = d-2Ppromoter-1 (4)

Thus, we obtain:

σDT = σ0 + (κEbPpromoter-1/2)d-1 = σ0 + kDTd-1 (5)

Eq. (5) answers the two key questions we raised at the beginning:

(a) DT has Hall-Petch like size scaling, with Hall-Petch exponent 1 by our simple model.

This arises because of the necessity to sustain perfect slip coherence in the thickness

direction. Forest dislocations of screw character can thread adjacent slip planes

together and promote stimulated slip, and the larger the area, the more likely the two

adjacent planes are effectively coupled, until a critical condition is reached where

perfect slip coherence can be sustained. As the limiting cases, DT always happens if d

is sufficiently large, even for small ρ and Ppromoter, and conversely, DT can be

suppressed altogether at small enough d, for any reasonable ρ and Ppromoter.

(b) kDT is so large because Ppromoter is small. κEb in (5) are the usual normalization



constants from ODP, and can be regarded as 1 in reduced units: in order for kDT >>

kODP by a factor of 10, as seen in many metals7, the magnitude of Ppromoter should be ~

10-2. This is reasonable, because only a small fraction of the stored forest dislocations

can serve as effective promoters for twinning slip in pole mechanisms7,12-14. By our

model, only 1% of the stored forest dislocations that penetrate the twinning system

glide planes are operative “helical staircases” that can promote stimulated slip on the

next plane.

Note that equation (5) comes from a very simple model. It is implicitly assumed that the

in-plane slip propagation is very easy, and coherence can be maintained in an entire d2 area

once started. Also, Ppromoter is assumed to be a constant with no stress dependence. Once

these complexities are taken into account, it is likely that the best-fit Hall-Petch exponent

could deviate from 1. Nonetheless, our simple model can already address the key questions

underlying the DT size dependence, and illustrate the nature of DT as a perfectly correlated,

stimulated slip phenomenon catalyzed by promoter defects.

Because of the large Hall-Petch slope kDT = κEbPpromoter-1/2, the Hall-Petch scaling for DT

σDT = σ0 + kDT d-1 would hit the ultra-strength regime 19,20 at a conspicuously large and

experimentally easily accessible d, and must therefore break down, because strain rate would

diverge to infinity when one approaches the ideal shear strength. For d < dc, the ODP

dislocations never manage to “catch fire and start lasing” (DT), because adjacent slip planes

are weakly effectively coupled due to small d2. In stress-ramp-up experiments, the stress σ

will manage to rise high enough to drive an ODP-mediated strain rate that matches the

applied strain rate, and reach steady-state flow, before it can reach the coherence transition.

The DT mechanical energy dissipation channel never gets activated, and the ODP dissipation

channel becomes the primary one throughout the loading path. In our analogy, the system is

then just an ordinary light bulb, not an optically-pumped laser11.

Finally, we would like to remark that the discussions above are based on a bulk promoter

population whose total number scales with the volume. DT proficiency may go up again over



a range of very small crystal sizes, such as 10 nm18, when the most effective promoters for

stimulated slip are GBs and surfaces instead of screw dislocations. This is beyond what we

have discussed in the paper. Based on an area-to-volume ratio argument, the promotion

efficacy of GBs/surfaces may exceed that of the bulk at very small crystal sizes, for emitting

the “right” sequence of partial dislocations for DT. Indeed, Chen et al. have argued that for

Al with high stacking-fault energy there can be some DT in very small nanograins, as the

partial dislocations get emitted more easily from the GBs at tiny grain sizes18. So the

observation of DT in nano-grains is not inconsistent with this paper, just at a very different

size regime. Also, unlike the bulk promoter case presented here, where DT dominance in

terms of contribution to the plastic strain at d > 1μm is clearly affirmed from the stress-strain

curves, in nanograined Al although deformation twins are observed in TEM, it is uncertain

that DT has contributed more to the net plastic strain than ODP or grain boundary sliding –

i.e. DT dominance was not affirmed as a regime in nanocrystalline Al.



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